In recent years there has been a great deal of interest in understanding the
fundamental behaviour of granular materials. Granular materials are ubiquitous in
natural and industrial settings however, their flow behaviour cannot be described
using classical ideas of fluid flows as they stand. Of particular interest are theories
which have been developed over the last 25 years. These ideas develop the analogy
between granular flows and the kinetic theory of gases, but unlike thermal fluids,
kinetic energy in granular systems is dissipated into heat during collisions and hence
is not conserved; one must balance the energy input rate with the dissipation rate due
to collisions in the system to achieve a steady state. Vibrofluidised granular beds are
often used as an idealisation of granular flows as they provide a convenient
approximation to the simplest type of flow: steady state, binary and instantaneous
collisions with no rotation. In this research work, we explore the behaviour of
vibrofluidised three-dimensional granular beds by developing various models based
on the granular kinetic theory approach. A finite element (FE) based software,
Comsol Multiphysics, was used as a toolkit to numerically compute solutions to the
three-dimensional conservation equations resulting from granular kinetic theory and
the results are shown in dimensionless units.
In the first case, an inviscid model for a vibrofluidised granular bed is
developed using only observable system parameters such as particle number, size,
mass and coefficients of restitution. Two closures based on granular kinetic theory are
described, one the standard Fourier law relating heat flux to temperature gradient, the
other including an additional concentration gradient term. Each prediction of the twodimensional
axisymmetric granular temperature and packing fraction fields was
compared against a previously validated one-dimensional model and threedimensional
experimental results, acquired using the technique of Positron Emission
Particle Tracking (PEPT). Both closures result in solutions that are in reasonable
agreement with the experimental results without any fitting parameters, but it was
found that differences between the predictions of each of the closures were relatively
small in comparison to the anisotropy of the experimentally determined temperature
distribution. The models resulting from both theories predict the existence of a small
non-zero radial pressure gradient due to a net radial force on any given volume
element in the cell, which is not balanced by the gravitational body force since gravity
acts parallel to the z axis.
Subsequently, considering the viscous effects on the system, a full NavierStokes
like viscous model was developed using the Standard Fourier type heat flux
based on granular kinetic theory. The resulting granular temperature and packing
fraction profiles compare well against the inviscid model and the PEPT experimental
results suggesting that the viscous effects are small. The mean velocity profiles from
the viscous model show the presence of asymmetric toroidal convection rolls in the
system that match well with the shape of the roll observed in the experiments.
Quantitatively, the mean velocity profiles show good agreement with the
experimental results at relatively low altitudes for a range of experimental values.
However, unlike the experimental results the viscous model results show trends in the
relationship between angular velocity at the centre of the convection roll and base
amplitude of vibration. Additionally, the wall effects are explored in the model which
shows that the convection rolls are influenced by the sidewall restitution coefficient, a
result that was earlier confirmed using the molecular dynamics simulations.
The viscous model was extended to predict the behaviour of an annular
vibrated three-dimensional granular bed. The results from the model are compared
with the molecular dynamics simulations and experimental data obtained using PEPT.
The predictions from the kinetic theory model for mean velocity, granular temperature
and packing fraction fields show good agreements despite the presence of anisotropy
in molecular dynamics simulations and experimental results. Subsequently the
particle-inner wall, particle-outer wall coefficients of restitution phase diagrams
generated from the model and the simulation results from molecular dynamics are
seen to be in excellent agreement. A comprehensive analysis to probe other key
factors that control the direction and magnitude of convection rolls was carried out.
This involved a study on five critical variables namely, the inner and outer wall
coefficients of restitution, number of grains, ratio of surface areas of the inner and
outer cylinders and base amplitude. The results from a systematic study indicate that
all the five variables examined can influence the direction and magnitude of the
convection rolls in the system. However, it is determined that to initiate convection
rolls the presence of energy dissipation at the walls is required. Finally, a comparison
between the double convection rolls previously observed experimentally and in
simulation shows excellent agreement suggesting that the model may further be used
to study the transition from single convection to double convection roll motion of the
grains and to explore the precise experimental conditions under which double rolls occur.

Description:

A Doctoral Thesis. Submitted in partial fulfillment of the requirements for the award of Doctor of Philosophy of Loughborough University.